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  1  Copyright © 2018 N. Rabino Analysis and Qualitative Effects of Large Breasts on Aerodynamic Performance and Wake of a “Miss Kobayashi’s Dragon Maid” Character  N.Rabino A RTICLE I  NFO  Keywords:  Computational fluid dynamics, ANSYS, drag coefficient, human aerodynamics, SST k- ω model, anime, Quetzalcoatl, titties, thicc  AMS Subject Classifications: 00A72, 76-05, 76G25 A BSTRACTA computational fluid dynamics methodology is used to study the salient flow features around the breasts of a human figure and to describe the aerodynamic differences imparted by their geometric presence. Two models are proposed for examination: a 3-dimensional reference based on a character design with a significantly buxom figure and a modification of this design where the breast size is reduced significantly. The two models are tested at speeds ranging from 1 to 30 m ⋅ s^-1 using Reynolds-averaged Navier Stokes (RANS). Drag, lift, and skin friction forces, along with turbulence kinetic energy (TKE), are investigated and compared between the differ-ent models. The present results are expected to provide useful information on the validity of the statement, “Flat is Justice” in terms of an aerodynamic standpoint. In addition to this, the results can offer worthwhile data investigating the anthropometrical presence of large breasts on sport aerodynamics. 1.Introduction The aerodynamics of the human form has been an area of valuableresearch in various aspects of sports and competition. Air resistance (hereinafter referred to as “drag”) is a concerning factor in many time- based trials, and enhancing potential efficiency can be done through the elucidation of the flow around the human figure. Studies concerning the drag of the human body using wind tunnels can be found dating back to the 1920s [1]. A small sampling of subsequent studies exploring the effect of drag covers areas such as running [2], cycling [3], skiing [4], and skating [5], all of which reinforces the relevance of aerodynamic investigation on the human shape in regards to performance. In many of such studies, the authors seek to investigate the effect of  positioning in relation to drag [6], and some utilize numerous subjects of differing anthropometric proportions to describe a generalized result on such positioning [7, 8]. Hitherto, none within the author’s investiga- tions has described the effect of specific physiological features on aero-dynamic performance in great detail. Stemming from certain internet communities and pertinent to the current era comes the succinct state-ment, “Flat is Justice”, which consequentially begets interesting debate that can reverberate and diffuse throughout media. Essentially, the statement describes the appreciation of flat-chested women [9], which  posits a peculiar aspect that has yet to be fully explored in human aero-dynamics; namely, the effect of breasts in regards to drag and overall aerodynamic performance. This work is intended to contribute to the understanding of how large breasts can affect the dynamics of the human wake through the use of computational fluid dynamics (CFD) simulation tools. This pre-liminary work focuses solely on comparing the relevant effects of large  breasts of a selected human design to that of the same design but with, euphemistically, “lesser tracts of land”. The following sections will  present an overall understanding on the human wake in relation to sim- plified geometry along with engineering applications, introduce the chosen human geometry and models, relevant boundary conditions, the governing equations, and the numerical methods used to solve the equa-tions. An in-depth review on the computational uncertainty is described, following with extensive results and discussion, conclusions, and rec-ommendations for future work. 1.1    Background on the Human Wake The human body can best be described as a bluff-body in respect to the flow around it. Literature on the behavior of the wakes behind bluff  bodies indicates that the flow will be unsteady due to the turbulent tran-sition and separation of the boundary layer [10]. A simplification analo-gous to the human shape can be represented by a grouping of uniform circular cylinders [11] and therefore existing studies on this type of geometry can provide general insight into the wake region. Sumner et al. [12] described the wake and development of vortex structures of cylinders with aspect ratios (i.e. height to diameter) of 3, 5, and 9, and determined that a transition in vortex shedding occurs at ℎ/ = 3 . An investigation by Okamoto and Sunabashiri [13] also supports this find-ing, adding that cylinders with an aspect ratio of 3 experience a recircu-lation region that extends four diameters downstream. Assuming the human form takes on a roughly large cylindrical shape near this aspect ratio, it is to be expected that the recirculation region will behave simi-larly and extend approximately four body widths downstream. A readily apparent deviation in geometry compared to studies done on singular cylinders is the presence of the gap between the legs. An extensive and comprehensive review done by Zhou and Alam [14] on the various arrangements of two cylinders indicate the wake structure falls into a multitude of regimes. In a side-by-side configuration, being similar to the two legs of a human, it is deduced that there are three  primary regimes where the wake experiences proximity interference. When closely spaced together, the first regime shows that the cylinders act similarly to that of a single bluff body with a width corresponding to the two cylinders. When the gap width is larger than 20% of the diame-ter, each cylinder has individual wakes that strongly affect one another and is associated with the second regime. At gap widths exceeding ap- proximately 100~120% of the diameter, each cylinder acts as an inde- pendent body with the vortex streets being loosely influenced by one another. Seeing that human legs are not strictly cylinders with a fixed diameter but more akin to inverted tapered cylinders, the wakes behind the legs will likely behave in a similar fashion observed in both the first and second regime. With the ankles and calves being narrower and hav-ing a larger gap between them, the second regime is applicable. A tran-sition into the first regime can be expected associated with the bulkiness of the thighs and reduction in gap width. Engineering literature can also provide additional details on the flow characteristics around the body. Many of such studies are motivat-ed by exposure control and contaminant transport [15, 16], thermal issues [17, 18], and comfort prediction [19], rather than overall drag effects. Inherently, many of the tested flow characteristics are evaluated in a quiescent environment or at air velocities that are of a lower order compared to those found in sport-related studies. Nonetheless, these studies provide useful insight on the natural turbulence caused by the human form and the expected anatomical location of flow separation. Inthavong et al. [20] utilized a high speed camera to record the wake generation of a 1/5th scaled realistic human manikin that was accelerat-ed to a velocity of ~1 m ⋅ s -1 . From their results, it was found that the    2  Copyright © 2018 N. Rabino shoulder undergoes flow separation and produces vortices in a regular  pattern. The hands produce a well-defined yet unstable vortex sheet that curls towards the centerline of the body. The head acts similarly to clas-sical sphere/cylinder cases with the addition of a trailing wake forming from behind the neck. The neck was found to remove the expected counter free shear layer that is present in cylinder studies and thus elim-inates the formation of an oscillating vortex sheet. In all, it can be said that the observed human wake is a highly complex and richly diverse system that is easily influenced by the inherent geometry used; it is expected that from this study, an overall summary can be presented on how and to what degree the previously described flow structures are affected by the presence of large breasts. 2.   Methodology 2.1    Design Proposal and Model Scaling The use of realistic human models affords greater realization of the  pertinent flow characteristics as they are considerably different than those of generalized models. Yan et al. [21] concluded that an excessive degree of simplification in using a manikin can affect the ability to achieve accurate results, and thus precludes the use of a simplified model for this study. However, the acquisition of a 3-D scanned human model with a significant bust indubitably proved difficult. The use of a highly unconventional approach was used to ameliorate this issue. The animated adaptation of  Miss Kobayashi’s Dragon Maid  , being a recently popular show [22] and spawning a sizable subculture on the internet [23], proved suitable in terms of providing potential models. The dragon characters (themselves being based off of mythically and culturally prominent dragons) assume a human form to interact with other humans in this well-received [24] slice-of-life urban fantasy. A majority of the human forms of the female dragon characters possessed significant busts. However, Quetzalcoatl (referred to canonically as “Lucoa” and will be named as such throughout the rest of this paper) substantiated herself as the adaptation’s gag character by her significant size [25, 26], thus making her the perfect candidate in providing a suita-  ble model. Being clearly the largest amongst her fellow dragons as es-tablished in Figure 1, Lucoa provides the best contrast between a large  bust and having none at all. To provide the most direct comparison in regards to the effect of large breasts on the wake, a dramatic reduction in bust size as reflected in Figure 1( b) was proposed for use in this study. In order to obtain accurate results from the setups described later in this paper, it is important to have the subject in question reflect real world scales properly. Lucoa’s height while in human form is not given explicitly in any related media within the scope of the author’s research. Thus, Lucoa’s height must be estimated in relation to objects of which a reasonable measurement can readily be found. Conveniently, there is a scene found within Episode 6, Season 1 of the animated adaptation wherein Lucoa steps through a doorframe. Assuming the door is of a typical size 1  used for external entrances, in addition to Lucoa being scaled properly in the scene, we can estimate her height using a vanish-ing point technique. Using the door as depicted in Figure 2 to judge Lucoa’s height, it was determined that she stands approximately 177 cm measured to the top of her hat, with her horns boosting her overall figure to a height of 182 cm. These numbers can be considered reasonable based on canoni-cal descriptions of Lucoa’s towering stature [27] compared to the aver-age height of 158 cm for a Japanese woman [28]. 2.2   3-Dimensional Models and Geometry Analysis Since Lucoa is a fictional character that is commonly portrayed in a 2-dimensional 2  world, determining her form drag between the two  proposed designs as described in Figure 1 requires that we add another dimension to her model. Conveniently enough, an available 3-D model of Lucoa [29] was used that would make the simulation possible. This  MikuMikuDance  3-D model (henceforth referred to as the “Normal” model) was then imported into the 3-D modeling program Blender, scaled to the determined height as described in the previous section, then exported into an STL file. This STL file was then repaired using the built-in repair feature present in Microsoft 3D Builder due to the unclean geometry inherent with the model. To achieve the modified design (henceforth referred to as the “Flat” model), the srcinal  Miku- MikuDance  model was modified using the built-in tools in Blender to dramatically reduce Lucoa’s breast size. The export and repair process remained the same as for the srcinal model. As shown in Figure 3, all positions between the two models remain the same and left unperturbed to leave the reduction in breast size as the sole geometric difference to be investigated. Although the typical or-thostatic (standing) orientation of a human has the upper limbs in a 1  A typical metric external door’s size is 926 mm wide by 2040 mm tall. 2  Referring to the media she is portrayed in, such as printed materials and televi-sion. (a) Original reference design. Courtesy: Kyoto Animation .   (b) The modified design proposed for comparison.   Figure 1. Comparison of different designs for Lucoa. Figure 2. A perspective measurement of Lucoa in reference to a door frame using the Vanishing Point Tool in Adobe Photoshop.  3  Copyright © 2018 N. Rabino relaxed position [30], the arms are left posed at a 45° adduction angle from the torso, as this is the default ‘A’ pose when importing the model. This arm position also has an advantage in this study as it potentially enables a more thorough analysis on the effect of breasts on the wake region, whereas a neutral standing posture would have the arms inter-fere with the downstream effect of the breasts. The hair is left modeled as solid to reduce simulation complexity and setup. 3  While humans naturally lean forward against the direction of the wind to maintain equilibrium [31], this factor is not considered in this study as this lean-ing would change the frontal area exposed to the fluid flow and thus complicate comparisons against static reference models. Dimensionally, the bounds of the two models are similar, with the height and arm span being 1.82 and 1.387 meters respectively. The  Normal model has a depth of 0.525 meters whereas the Flat model is only 0.414 meters. The frontal projected area,  , of both models is 0.584 m 2 . The volumetric difference between the two is 9.19 L, indicat-ing that each breast on the Normal model has an enormous volume of approximately 4.6 L. The under-bust circumference of the Normal mod-el is approximately 64 cm and the bust measures 115 cm. The Flat mod-el has the same under-bust measurement whereas the bust measures 68 cm.Attempting to match the dimensions and bust volume of the Normalmodel to existing cup sizing scales is difficult as these measurementsare exceptionally large and exceed volumes measured in other studies[32]. Using the JIS L 4006:1998 [33] scale and extrapolating 4  cup siz-ing from the largest listed size (I-cup), the Normal model can be de-scribed as being 10 cups larger; an estimated “S65”. The Flat model is astark contrast to this, where it matches a petite “AA65” size.The dramatic difference in bust size between the models serves to  provide the most significant change in outcomes; it is assumed that due to the absurd bust size, any size smaller than the Normal model would have an outcome that would fall in a range between both models. 2.3    Evaluated Metrics and Implementation Four metrics under investigation for this study include drag and lift forces (including their associated coefficients), skin friction coefficient, and finally, turbulence kinetic energy. To evaluate the drag coefficient, C D , and drag force, F D , the following equations are used, C D  = 2F D   ∞2   (1) F D  = (− cos +    sin)   (2) 3  Hair physics is beyond the scope of the author, and thus this study, due to the inordinate amount of computing resources and time needed to setup and simulate hair strands in a physically accurate fashion. 4  In [33], each cup size is binned with every 2.5 cm deviation from the under- bust measurement starting from 7.5 cm. where   is the fluid density,   ∞  is the free stream velocity,   is the frontal projected area, and    is the pressure at the surface  .     is the local wall shear stress being defined as,     ≡    =0  (3) with   as the dynamic viscosity,   the flow velocity along the boundary, and   being the height above the boundary. The value of C D  is not con-stant and is dependent on Reynolds number, which is defined as, Re  = 󝠵  (4) where 󝠵  is an arbitrary characteristic length. In this study, 󝠵  is equal to the height of the models. The lift coefficient is comparable to the drag coefficient, being that the force is evaluated in a direction that is perpendicular to the mean flow direction, e.g. vertically upwards. Thus, C L  = 2F L   ∞2  􍠵  (5) F L  = (− sin +    cos)   (6) Instead of the frontal projected area,  , a reference surface area, 􍠵  , is used. For consistent comparison however,   and 􍠵   are left defined as  being equivalent, thus  = 􍠵  . This result does not affect the calculated forces but rather only the coefficient, and as such, the lift coefficient is dependent on the frontal area. The skin friction coefficient, C f  , is evaluated in a similar manner to the drag coefficient since the force attributed to skin friction is a com- ponent of the profile drag, F D . Therefore, C f   = 2τ    ∞2 (7) Analyzing the skin friction coefficient allows insight into areas where the boundary layer thickness changes; as turbulent flow increases, the thickness of the boundary layer increases, and consequently areas where C f   transitions to larger values or experiences spikes are indicative of where flow separation is prevalent [34, 35]. Turbulence kinetic energy (TKE) signifies of the loss of kinetic en-ergy from the mean flow and represents the energy present with eddies in turbulent flow; it is a direct measure of the intensity of turbulence. In a general form quantifying the mean of turbulence normal stresses, TKE is defined as, 󽠵 = 12(′) 2 󟿽 + (′) 2 󟿽 + (′) 2 􏿽󿿽  (8) The exact value of TKE is calculated based on the closure of the Reyn-olds-averaged Navier-Stokes equations, which is further discussed in Section 3.3.  The numerical simulations in this present work, along with the au-tomatic evaluation of the equations described in this section, were car-ried out using ANSYS FLUENT R17. The 3-D models defined in Sec-tion 2.2 were imported into FLUENT and followed the methodology as described in the following section. 3.Computational Fluid Dynamics (CFD) Setupand Analysis 3.1    Boundary Conditions The use of boundary conditions based on real-world environments enhances the overall applicability of the results stemming from the sim-ulations. It was therefore important to determine the most appropriate and accurate environment in which to simulate the models with. It was (a)Reference (Normal) model.(b)Modified (Flat) model. Figure 3. 3-D representations of Lucoa to be used in CFD simulations, detailing (clockwise) top, side, and front views.  4  Copyright © 2018 N. Rabino found that the overall location used in the animated adaptation of  Miss  Kobayashi’s Dragon Maid   was based on the city of Koshigaya [36], situated in the Saitama Prefecture of Japan. A logical time of year to assume a person being outside without excess clothing would be some-time in the summer. Using the month of August, it was found that weather conditions in Koshigaya and nearby surrounding regions fea-ture averages [37] of 22.6°C for temperature, 73% for relative humidity, and 1005.9 hPa for local atmospheric pressure. Thus, the air density was calculated to be  = 1.1581 kg ⋅ m −3  and the dynamic viscosity to be  = 1.86847 ⋅ 10 −5  kg ⋅ m −1  ⋅ s −1 . Since the human body can vary based upon the clothing worn, sur-face roughness and the effects of fabrics are parameters that are ignored in this study. Although multiple studies have shown fabrics have a no-ticeable effect on the overall drag of a human body [7, 38], the walls in this computational work can be regarded as smooth. In all simulations, the models and ground of the domain are modeled as non-moving walls with no-slip conditions. The clothing that is part of the models is treated in the same manner. Table 1.  Summary of boundary conditions in the present study. Wind speed   ∞  m ⋅ s −1  1.0, 2.5, 5.0, 7.5, 10.0, 15.0, 20.0, 25.0, 30.0 Reynolds  Number Re   −  1.281 ⋅  10 5  ~ 3.384 ⋅  10 6 Domain bounds −   m   �−1.44 ≤  ≤ 1.44−2.34 ≤  ≤ 5.650 ≤  ≤ 2.32  Turbulent intensity   inlet  −  1%   outlet,backflow  −  5% Turbulent viscosity ratio    inlet   −  10    outlet,backflow   −  10 Outlet gauge  pressure     Pa  0 Inlet velocities range from 1 m ⋅ s -1  to 30 m ⋅ s -1  in the positive y-direction (since in this respect, the positive z-direction refers to the “up” orientation; refer to Figure 4 for clarification), highlighting typical wind speeds encountered on a day-to-day basis such as walking [39] all the way up to standing in a violent storm [40]. At the inlet, turbulence is specified using both turbulence intensity,   , and turbulent viscosity ratio,    / . Turbulence intensity is defined as the ratio of the root-mean-square of velocity fluctuations, ′ , to the mean flow velocity,   ∞ , and the turbulent viscosity ratio being directly proportional to the turbulent Reynolds number (    ≡ 󽠵 2  / ). These values are summarized in Table 1. The boundary condition at the outlet is treated as a pressure outlet where a static gauge pressure is specified. In this case, turbulence is specified similarly as the inlet but regarded in terms of “backflow”, should the flow reverse direction at the boundary during iterative calcu-lations. The remaining borders of the “virtual wind tunnel” are modeled as symmetric to simulate zero-shear slip walls. In FLUENT, this bound-ary condition assumes a zero flux for all quantities, which imposes a zero normal gradient across the defined boundary and thus enforces a  parallel flow. In FLUENT, the flow is initialized with a velocity field equal to the specified velocity for the run, e.g., a run specified at 1.0 m ⋅ s -1  would have the entire field initialized with that value, and so on. Turbulence  parameters at the boundaries are also initialized based on turbulence values as specified in Table 1. The blockage ratio was determined to be 8.7%, which would necessitate the usage of a correction factor to data; however, a blockage ratio of up to 10% in regards to bluff bodies has shown to provide reasonably similar outcomes compared to testing using lower blockage ratios [41] and therefore a correction factor was not used.   3.2   Grid Generation The computational domain was discretized with an unstructured grid as shown in Figure 5. To reduce numerical diffusion and to more accurately resolve the viscous boundary layer, the surface grids on the models and ground were extruded using prismatic elements that are sized appropriately to the aspect ratio of their associated surface cell. These prisms are grown to 5 layers and follows recommendations put forth by Lanfrit [42]. Two prismatic bodies of influence (BOI) of in-creasing refinement are used to improve the resolution of the grid in  both the wake region and the surrounding area around the models to sufficiently capture turbulence and flow separation. This is done to en-sure that computational processing is focused on more important re-gions in the flow regime while keeping the far field sufficiently coarse enough as to not dramatically hamper computational time. The overall grid is limited to a maximum spacing of 0.1 m and a minimum of 2 mm. The smaller, finer BOI is sized by the bounds −0.894 ≤  ≤ 0.894 , −0.1 ≤  ≤ 2.76 , 0 ≤  ≤ 1.87  and the larger, coarser BOI defined by −0.944 ≤  ≤ 0.944 , −1 ≤  ≤ 4.26 , 0 ≤  ≤ 1.92 , all in meters. A conversion algorithm in FLUENT was used to convert the pre-liminary tetrahedral and prismatic grid into a polyhedral one. Polyhedra exhibit advantages over tetrahedra, namely, they approximate gradients  better than tetrahedra due to the fact they are bounded by many neigh- bors. Additionally, polyhedra have more lax geometric criteria due to their insensitivity to stretching, making grid pre- and post-processing easier; this is well suited to the highly complex geometry of the models used. It has been observed that polyhedral grids provide the same level of accuracy as tetrahedral ones, but of significantly lower element count, thereby hastening simulations [43]. Furthermore, polyhedral grids have shown to improve convergence while having notably greater accuracy under unsteady simulations [44]. This is further supported by similar external aerodynamic studies run under FLUENT, where speedups between 2 to 3 times towards a converged solution have been observed [45]. 3.3   Turbulence Model and Computational Approach The flow around the models is modeled with Reynolds-averaged  Navier-Stokes (RANS) equations in incompressible form. Written in Cartesian tensor form and having flow variables of the form  =   ̅+ ′  (with   ̅  and ′  being the mean and fluctuating components respectively) Figure 4.  Boundary conditions of the computational domain, with the inlet being represented in blue, outlet in red, walls in white, and symmetry in yellow. Figure 5.  Side view of the full grid domain along the median plane.
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